Intriguing Lessons about Teaching and Assessing Math

around the World

Steve LeinwandAmerican Institutes for Research

SLeinwand@air.org

NCTM – April 28, 2012

What a great time to be convening as teachers of

mathematics!• Common Core State Standards Initiative• Quality K-8 instructional materials• More access to material via the web than ever• $5 billion with a STEM RttT tie-breaker• A president who believes in science and data• The beginning of the end to Algebra II• A long overdue understanding that it’s instruction

that really matters• A recognition that the U.S. doesn’t have all the

answers.2

And what can we learn from others?

• There is more than one way to do things.• Like how a foreign language strengthens

ones understanding of English, so too can international comparisons hold up mirrors.

• There might be good reasons why other nations outperform the U.S.

• In answering “How is it done elsewhere?” we can learn how to change what we do here.

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Today’s agenda

Glimpses of intriguing differences in:• Curriculum Frameworks and Standards

• Instruction

• Assessment

• Teacher Support

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Part 1

Curriculum Frameworks and Standards

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How NCTM (and most states) Organizes the Curriculum

• Five Content Strands– Numbers and

operations– Measurement– Geometry– Algebra– Data and

Probability

• Five Process Strands– Problem-Solving– Reasoning and

Proof– Communication– Connections– Representations

How Singapore Organizes the Curriculum

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Singapore logically builds-up math topics across grades: numbers

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A composite of Singapore, HK, Korea

Gr. 2 Gr. 3 Gr. 4 Gr. 5

Fractions / concepts

Fraction of a whole: Interpret a fraction as part of a wholeRead and write fractionsCompare and order unit fractions and like fractions.(denominators less than or equal to 12)

Equivalent fractions:Recognize and name equivalent fractionsWrite the equivalent fraction of a fraction, given the denominator or the numeratorExpress a fraction in its simplest formCompare and order unlike fractions, including comparing fractions with respect to one half(denominators less than or equal to 12)

Mixed numbers and improper fractions:Understand the concepts of mixed numbers and improper fractionsExpress an improper fraction as a mixed number, and vice versa, and expressing both in simplest form

Fractions / arithmetic operations

Addition and subtraction of two related fractions (one denominator a factor of the other) within one whole (denominators of given fractions should not exceed 12)

Addition and subtraction of- like fractions- related fractions (denominators of given fractions should not exceed 12)

Addition and subtraction of fractions with unlike denominators:Add and subtract fractions with unlike denominators

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A composite of Singapore, HK, Korea

Gr. 3 Gr. 4 Gr. 5 Gr.6

Perimeter Develop the concept of perimeter

Measure and find the perimeter of 2‑dimensional shapes

Find one dimension of a rectangle given the other dimension and area and perimeter

Understand the area and the circumference of circles

Understand the concept of pi

Find the area and the circumference of circles, semicircles, and quarter circles

Area Develop the concept of area

Compare areas, using improvised units

Measure area in square centimeters (cm2) and square meters (m2)

Apply the formula for area of squares and rectangles and

composite figures made up of rectangles and squares

Apply the formula for the area of triangles, parallelograms, and rhombuses

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K-6 Density or A Glimpse at a Mile Wide

Total No. of Topics

Avg. No. of Topics per

Grade

Avg. No. of Grades/Topic

Avg. No. of Outcomes/Grade

(1) (2) (3) (4)

Singapore 40 15 2.3 39

California 42 202.9

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Florida 54 39 4.2 107

Maryland 46 29 3.8 69

New Jersey 50 28 3.4 56

N. Carolina 41 18 2.6 36

Ohio 48 26 3.3 62

Texas 40 19 2.8 44

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Singapore, Japanese and Hong Kong Math Standards at: http://hrd.apec.org

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Basic rationales for the new Chinese math standards:

• We cannot do without mathematics in our daily living, work and study.• Contents of mathematics learning for school children ought to be realistic, meaningful and challenging.• Mathematical instructional activities should be based on children’s cognitive developmental level and build on past experiences.• Modern information technology has great impact on the values, objectives, contents and pedagogy of mathematics education.

A peek at China’s approach:• What is the approximate thickness of 1200 sheets of

paper? What is the approximate number of classes that may be formed comprising of 1200 students? What is the approximate length of 1200 footsteps? (grades 1-3)

• Estimate the number of words (or characters) contained in one whole page of a newspaper. (grades 4-6)

A 2nd peek at China’s approach:(grades 10-12)

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Initial Lessons

• Think holistically, and systemically, not linearly and piecemeal

• Coherence and focus• Learning progressions• “Teach less, learn more”• Provide examples

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Part 2

Instruction

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A Glimpse at Two Grade 1 Textbooks

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Textbook Grade 1

#Topics

# Les-sons

Average Pages/Lesson

Pages of Develop-

ment

Pages of Exer-cises

Other Pages

Sing-apore

13 34 15174

(35%)261

(53%)62

(12%)

U.S. (Scott Fores-man)

25 157 4145

(26%)169

(30%)250

(44%)

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How about content balance at grade 1?

Textbook Number Measurement Geometry Algebra Data

Singapore 27 (79%)5

(15%)1

(3%)0 1 (3%)

Scott 106 (68%)27

(17%)11

(7%)5 (3%) 8 (5%)

Everyday Math* 62 (56%)

17 (15%)

8 (7%)

2 (2%) 7 (6%)

*14 (13%) review and routines lessons

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A Glimpse at Three Grade 6 Textbooks

Textbook#

Chapters#

LessonsTotal Pages

Average Pages/Lesson

Pages of Development

Pages of Exercises

Other Pages

Singapore 11 24 402 17107

(27%)132

(33%)163

(41%)

Scott 12 158 740 5202

(27%)216

(29%)322

(44%)

Everyday Math 10 113 411 4

299 (73%)

112 (27%)

0

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How about content balance at grade 6?

Textbook Number Measurement Geometry Algebra Data

Singapore 15 (63%)4

(17%)3

(13%)1

(4%)1

(4%)

Scott 64 (41%)14

(9%)15

(9%)18 (11%) 17 (11%)

Everyday Math* 37 (33%)

5 (4%)

18 (16%)

19 (17%) 23 (20%)

*11 (10%) review lessons

But what about depth, rigor and connections?

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2323

Cost of Raising a Child to Age 18 (for each $100)

U.S. Textbook Problems Emphasize Mechanical Formulas:

Gr. 6 Pie Chart Requires Summing to a Total

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Singapore textbooks uses structured variation and multi-step Problems :

Gr. 6 Pie Chart Problem Incorporating Angles

And then there is Singapore’s visual approach to mathematics:

The bar model

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http://www.thesingaporemaths.com/

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And if we crank it up a bit….

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And 25! slides later:

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But what about the actual art of teaching?

Here’s a glimpse at the Japanese 6th grade lesson.

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The problem- Grade 6

We bought pencils and ballpoint pens and the total number of items was 10 and the price was 460 yen. The price of each pencil was 40 yen and the price of each ballpoint pen was 70 yen.

How many pencils and how many ballpoint pens did we buy?

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# of pencils

0 1 2 3 4 5 6 7 8 9 10

# of pens 10 9 8 7 6 5 4 3 2 1 0

Total cost

700 670 640 610 580 550 520 490 460 430 400

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Let’s take a peek

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Part 3

Assessments

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A Question of Balance

% Multiple Choice

% Multi-step

% Finding an Intermediate

Unknown

Singapore – 6 31 25 19

Texas – 8 100 6 2

Mass – 8 77 13 5

Ohio – 8 74 17

NAEP – 4 64 15 4

NAEP - 8 60 21 8

Massachusetts – Grade 3 – 2007

Alan has the number tiles shown below. 4 7 8 1• Use all of Alan’s number tiles to make the four digit number ‑

with the smallest value. Use each number tile only one time. Write the number in the boxes below.

• What is the value of the digit 7 in the number you made? Explain your answer.

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Hong Kong – Grade 3 - 2007

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Hong Kong – Grade 3 - 2007

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Hong Kong – Grade 3 - 2007

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Hong Kong – Grade 3 - 2007

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Mass vs. Hong Kong – Grade 3

Characteristic

Massachusetts Hong Kong

MCAS Spring 2007 Released Items 2007 Territory‑Wide Spring System Assessment

Items by strand** Number:            17 (49%)

Measurement:   4 (11%)

Geometry:         4 (11%)

Data:                6 (17%)

Algebra:            4 (11%)

Number:            15 (42%)

Measurement:   12 (33%)

Geometry:         7 (19%)

Data:                2 (6%)

Algebra:            0 (0%)

Multiple‑choice items

                        25 (71%)                         5 (14%)

Constructed‑response items

10 constructed response(all but two are short closed

constructed‑response)

31 constructed response(all but two are short closed

constructed‑response)

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Mass vs. Hong Kong – Grade 3

Characteristic

Massachusetts Hong Kong

MCAS Spring 2007 Released Items

2007 Territory‑Wide Spring System Assessment

Items with graphics 20 (57%) 25 (69%)

Items within real‑world contexts

19 (54%) 21 (58%)

Items by computational difficulty

Low:                 34 (97%);

Medium:           1 (3%);

High:                0 (0%)

Average (1, 2, 3) = 1.03

Low:                 22 (61%);

Medium:           14 (39%);

High:                0  (0%)

Average (1, 2, 3) = 1.39

Items by cognitive complexity***

Level 1:             23 (66%)

Level 1+:           0

Level 2:             12 (34%)

Level 2+:           0

Level 3:             0

Average: 1.34

Level 1:             12 (33%)

Level 1+:           4 (11%)

Level 2:             16 (44%)

Level 2+:           3 (8%)

Level 3:             1 (3%)

Average: 1.6850

A Singapore Grade 6 Hard Problem You Won’t See on U.S. Grade 8 State Assessments

Source: Singapore MOE

Part 4

Teacher Support

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Primary education teacher Salaries Ratio of salary after 15 years of experience to GDP per capita (2004)

Singapore primary teacher preparation: a focus on pedagogical content

• Teaching and Learning of Primary Mathematics I (Numbers)

Overview of the Singapore Primary Mathematics Curriculum; Preparation of Scheme of Work and Lesson Plans; Pedagogical Strategies and Psychological Theories; Teaching of Whole Numbers, Fractions, Decimals, Percentage, Ratio and Direct Proportion, Rate and Speed. [Common pupils’ errors will be dealt with in the teaching of various topics.]

• Teaching and Learning of Primary Mathematics I I (geometry, measurement, statistics, algebra)

Teaching Problem Solving and Investigations; Mathematical Communication; Teaching of Geometry, Money and Measures, Mensuration, Graphical Representation and Statistics, Algebra. [Common pupils’ errors will be dealt with in the teaching of various topics.]

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Singapore primary teacher preparation: a focus on pedagogical content (cont)

• Teaching and Learning of Primary Mathematics III (e.g. assessment)

This course covers two broad areas: (a) Various traditional assessment modes in Mathematics and the use of these modes in schools to assess pupil performance, in particular, the planning and construction of test items and (b) Practice of teaching skills, including catering for pupils of mixed abilities.

• Teaching and Learning of Primary Mathematics IV (e.g., IT)

This course provides further pedagogical skills for those who will go deeper into teaching mathematics, especially at upper primary levels. Topics include: Games in Mathematics; Challenging Problems in Upper Primary Mathematics. Student teachers will also undertake Independent Study Topics which will enhance their teaching repertoire.

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Professional development:Japanese lesson study is an application of quality principles to improve teaching

1. Identify a teaching topic based on student needs2. Plan a lesson with student learning as a focus 3. Teach a lesson with Japanese form (identify math

concept, present problems, discuss solutions, summarize what is learned)

4. Evaluate lesson’s impact (with others educators)5. Revise the lesson

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China’s elementary specialists• Chinese elementary teachers specialize by

teaching either mathematics or language but not both.

• Most common model is teachers teach mathematics grades 1-3 or 4-6.

• Teachers move from classroom to classroom and young students stay in same room.

• Teachers don’t have their own home room, they share a common teacher room promoting teacher interaction.

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The teaching hierarchy in Chinese cities

• Trainee teacher (3 years)• Second rank teacher (if you qualify) – can

teach independently• First rank teacher - need to write articlesThen in Beijing or Shanghai: teach/mentor by:• Master or senior teacher (2%)• Leading or special teacher (0.4%)• Super rank (te ji jiao shi) (0.15%) via contests!

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Lots to think about.Lots to consider emulating.

Lots of ways we can be more effective.

Thank you.SLeinwand@air.org

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A Virtual Handout• Lesson Study Plan and Video:

http://hrd.apecwiki.org/index.php/Classroom_Videos_from_Lesson_Study

• What the US can Learn from Singapore: http://www.air.org/files/Singapore_Report_Bookmark_Version1.pdf

• MA vs. Hong Kong: http://www.air.org/files/AIR_Measuring_Up_Report_0427091.pdf

• APEC Standards: http://hrd.apec.org • Singapore Bar Model: http://www.thesingaporemaths.com

• Steve Leinwand: SLeinwand@air.org60

Or Google:

• Apec wiki lesson study• What the U.S. can Learn from Singapore• Massachusetts vs. Hong Kong + measuring up• The Singapore Maths.com

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